Question 1

This question concerns the volume of an object bounded by a parabola x = y² - 5y on the left and a

straight line x = y on the right (z and y in metres). The cross section is show below:

Ty

x = y? — 5y

The height of the object, h(x, y) (in metres), is modelled in this assessment using the following function:

h(x,y) = (y − z)(z − y? +5y).

x=y

(a) (5 marks) Find the maximum height by doing the following:

i) find all the critical points of the function h(x, y),

ii) identify the single critical point that is within the cross section area A depicted above (not

including points on the boundary).

iii) show that this critical point is a local maximum using the Hessian determinant test,

iv) evaluate the value of h at this maximum

(b) (5 marks) The volume of the object V (metres³) is defined as the integral

=ff h(x, y) da

V =

where A is the cross-section depicted in the figure above. Calculate the volume by doing the

following:/n(c) (2 marks) use MATLAB to create a contour plot of the height h(x, y). Make sure to include a few

(three or more) contours (level curves) between zero and the maximum value you found in Qla.